Problem 139 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the equation represents \(y\) as a function of \(x .\) $$x^{2}+y=5$$

Step-by-Step Solution

Answer

Yes, the equation \(x^{2}+y=5\) represents \(y\) as a function of \(x\), as for every \(x\) value, there is one unique \(y\) value.

Step 1: Rewriting the equation

Let's isolate \(y\) on one side, so it would be easier to see if for every \(x\) there is one and only one \(y\). To do so, subtract \(x^{2}\) from both sides of the equation. This gives: \(y=5-x^{2}\).

Step 2: Analyzing if y is a function of x

From Step 1, it is clear that for every value of \(x\), there is only one corresponding value of \(y\). This means that \(y\) is a function of \(x\), as there is only one unique \(y\) value for every \(x\) value.

Key Concepts

Relation Between x and y

In mathematics, understanding the relationship between two variables, like \(x\) and \(y\), is crucial when analyzing functions. A function describes a specific kind of relationship where every input (typically \(x\)) is related to exactly one output (typically \(y\)).
For the equation \(x^2 + y = 5\), we want to determine if \(y\) is a function of \(x\). This means we need to check if each value of \(x\) maps to a unique value of \(y\).
If we can express \(y\) in terms of \(x\) in a way that doesn't result in multiple \(y\) values for a single \(x\) value, the relation is a function. In simpler terms, plug an \(x\) value into the equation and see if you always get a single answer for \(y\).

Isolating Variables

Isolating a variable means rearranging an equation to express one variable explicitly in terms of others.
This process is helpful when checking if an equation defines one variable as a function of another.
In our example, the goal is to solve for \(y\). Start with the original equation:

• \(x^2 + y = 5\)

Subtract \(x^2\) from both sides:

• \(y = 5 - x^2\)

Now, \(y\) is isolated. This tells us that \(y\) depends directly on \(x\). For each \(x\), there is only one \(y\). Consequently, the isolated equation clearly shows that \(y\) is determined by the equation, confirming the function relationship.

Unique Solutions in Algebra

When determining if an equation represents \(y\) as a function of \(x\), focus on whether every input yields a unique output. This forms the essence of unique solutions in algebra.
In our example after isolating \(y\), the equation \(y = 5 - x^2\) shows that for each \(x\) value, there is exactly one \(y\) value.
This uniqueness criterion is central:

• If for any \(x\) there were two different \(y\) values, \(y\) wouldn't qualify as a function of \(x\).
• Here, the squared term \(x^2\) doesn’t cause \(y\) to have multiple outcomes for the same \(x\).

Understanding unique solutions ensures clarity on function determination, vital for solving algebraic problems accurately.